Ncert Ex 3.1 Class 10

NCERT Ex 3.1 Class 10: A complete walkthrough to Understanding Pair of Linear Equations in Two Variables

This article provides a detailed explanation and solution to each problem in NCERT Ex 3.1, Class 10, focusing on the concept of pair of linear equations in two variables. Understanding this chapter is crucial for building a strong foundation in algebra and tackling more complex mathematical problems in the future. In real terms, we'll explore various methods of solving these equations, including graphical representation, and dig into the underlying mathematical principles. This guide is designed to be accessible to students of all levels, offering clear explanations and step-by-step solutions.

Introduction to Pair of Linear Equations in Two Variables

A linear equation in two variables is an equation that can be written in the form ax + by + c = 0, where a, b, and c are constants, and x and y are variables. A pair of linear equations in two variables consists of two such equations. The solutions to these equations represent points where the lines represented by the equations intersect on a coordinate plane.

The solutions can be:

Unique solution: The lines intersect at exactly one point.
Infinitely many solutions: The lines are coincident (they overlap).
No solution: The lines are parallel and never intersect.

This chapter primarily focuses on finding unique solutions using various methods, which we'll explore in detail while solving NCERT Ex 3.1 problems.

NCERT Ex 3.1: Detailed Solutions and Explanations

Let's go through each question in NCERT Ex 3.1, Class 10, providing comprehensive solutions and explanations. Note that the exact wording and numbering of problems might vary slightly depending on the edition of the textbook.

Question 1: Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” (Isn’t this interesting?) Represent this situation algebraically and graphically.

This question involves formulating a pair of linear equations from a word problem. Let's represent Aftab's current age as 'x' and his daughter's current age as 'y'.

Seven years ago: Aftab's age was (x-7) and his daughter's age was (y-7). The given condition is: x - 7 = 7(y - 7) => x - 7y = -42
Three years from now: Aftab's age will be (x+3) and his daughter's age will be (y+3). The given condition is: x + 3 = 3(y + 3) => x - 3y = 6

We now have a pair of linear equations:

x - 7y = -42
x - 3y = 6

To represent this graphically, we need to find at least two points for each equation. Let's solve for x and y in each equation:

Equation 1: x = 7y - 42

If y = 0, x = -42
If y = 6, x = 0

Equation 2: x = 3y + 6

If y = 0, x = 6
If y = -2, x = 0

Plot these points on a graph and draw the lines. The point of intersection of these two lines represents the solution – Aftab's and his daughter's current ages.

Question 2: The coach of a cricket team buys 3 bats and 6 balls for Rs.3900. Later, she buys another bat and 3 more balls of the same kind for Rs.1300. Represent this situation algebraically and geometrically.

Let's represent the cost of a bat as 'x' and the cost of a ball as 'y'.

First purchase: 3x + 6y = 3900
Second purchase: x + 3y = 1300

We have a pair of linear equations. To represent this geometrically, we again need to find points for each equation:

Equation 1: 3x + 6y = 3900 => x = (3900 - 6y)/3 = 1300 - 2y

If y = 0, x = 1300
If y = 650, x = 0

Equation 2: x + 3y = 1300 => x = 1300 - 3y

If y = 0, x = 1300
If y = 1300/3, x = 0

Plot these points and draw the lines. The intersection point gives the cost of one bat and one ball.

Question 3: The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.

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Let the length of the rectangle be 'l' and the breadth be 'b'. The area is lb.

Scenario 1: (l-5)(b+3) = lb - 9
Scenario 2: (l+3)(b+2) = lb + 67

Expanding these equations, we get:

lb + 3l - 5b - 15 = lb - 9 => 3l - 5b = 6
lb + 2l + 3b + 6 = lb + 67 => 2l + 3b = 61

Now we have a pair of linear equations to solve for 'l' and 'b'. You can solve this using substitution or elimination methods.

Question 4: On comparing the ratios a1/a2, b1/b2 and c1/c2, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:

This question tests your understanding of the conditions for unique solution, no solution, and infinitely many solutions. The given pairs of equations are:

(i) 5x - 4y + 8 = 0; 7x + 6y - 9 = 0 (ii) 9x + 3y + 12 = 0; 18x + 6y + 24 = 0 (iii) 6x - 3y + 10 = 0; 2x - y + 9 = 0

Compare the ratios a1/a2, b1/b2, and c1/c2 for each pair:

If a1/a2 ≠ b1/b2, the lines intersect at a point (unique solution).
If a1/a2 = b1/b2 ≠ c1/c2, the lines are parallel (no solution).
If a1/a2 = b1/b2 = c1/c2, the lines are coincident (infinitely many solutions).

Apply these conditions to determine the nature of the lines for each given pair of equations.

Question 5: Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

Let the width be 'w' and the length be 'l'. We are given that l = w + 4. The perimeter is 2(l+w). Half the perimeter is l + w = 36.

Substitute l = w + 4 into the second equation: (w+4) + w = 36 => 2w + 4 = 36 => 2w = 32 => w = 16

That's why, l = w + 4 = 16 + 4 = 20

The dimensions are length = 20m and width = 16m.

Solving Linear Equations: Methods and Techniques

NCERT Ex 3.1 primarily focuses on the graphical method, but don't forget to understand other methods for solving pairs of linear equations:

Graphical Method: Plot the lines on a graph. The point of intersection represents the solution.
Substitution Method: Solve one equation for one variable and substitute it into the other equation.
Elimination Method: Multiply equations by constants to eliminate one variable and then solve for the other.
Cross-Multiplication Method: A shortcut method derived from the elimination method.

Frequently Asked Questions (FAQ)

Q: What is the significance of comparing ratios a1/a2, b1/b2, and c1/c2?

A: Comparing these ratios helps determine the nature of the lines represented by the equations, indicating whether they intersect, are parallel, or coincident. This directly relates to the number of solutions the system of equations has.

Q: Can I use a calculator to solve these problems?

A: While a calculator can help with arithmetic calculations, the emphasis in this exercise is on understanding the methods of solving linear equations and interpreting the results. Focus on the process, not just the final answer.

Q: What if I get a decimal answer for the variables?

A: Decimal answers are perfectly acceptable. The important part is to accurately apply the methods to arrive at the solution.

Q: How can I improve my understanding of this chapter?

A: Practice is key. On the flip side, try solving more problems from textbooks and online resources. Understanding the underlying concepts is crucial; don't just memorize the steps.

Conclusion

NCERT Ex 3.By understanding these concepts and practicing regularly, you'll build a strong foundation for more advanced topics in algebra. Plus, mastering this chapter requires not only solving the problems but also understanding the underlying concepts of linear equations, their graphical representation, and the various methods of finding solutions. So naturally, 1 Class 10 provides a foundational understanding of pairs of linear equations in two variables. Remember to focus on understanding the process and applying different methods to develop your problem-solving skills. This thorough understanding will be essential as you progress to more complex mathematical concepts in the future.

Ncert Ex 3.1 Class 10